Daryl Cooper, Genevieve S Walsh
Suppose that M is a fibered three-manifold whose fiber is a surface of positive genus with one boundary component. Assume that M is not a semi-bundle. We show that infinitely many fillings of M along dM are virtually Haken. It follows that infinitely many Dehn-surgeries of any non-trivial knot in the three-sphere are virtually Haken.
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We use Heegaard splittings to give some examples of virtually Haken 3-manifolds.
In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [CG] and extends [MMZ].
We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be non-fibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to the 2 generator, non-fibered knot manifolds in the cusped Snappea census. For each such manifold M, we compute a number c(M), such that, for any n>c(M), the n-fold cyclic cover of M is large.
We describe a class $\mathcal{C}$ of punctured torus bundles such that, for each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are virtually Haken. We show that $\mathcal{C}$ contains infinitely many commensurability classes, and we give evidence that $\mathcal{C}$ includes representatives of ``most'' commensurability classes of punctured torus bundles. In particular, we define an integer-valued complexity function on monodromies $f$ (essentially the length of the LR-factorization of $f_*$ in $PSL_2(\mathbb{Z})$), and use a computer to show that if the monodromy of $M$ has co…
A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the complete Hodgson-Weeks census of 10,986 small-volume closed hyperbolic 3-manifolds, and for each of them found finite covers which are Haken. There are interesting and unexpla…
We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard splittings. We first give a necessary and sufficient condition, in terms of spectral geometry, for a finitely presented group to have a finite index subgroup with infinite abelianisation. For negatively curved 3-manifolds, we show that this is equivalent to a statement about generalised Heegaard splitt…
Symplectic fillings of standard tight contact structures on lens spaces are understood and classified. The situation is different if one considers non-standard tight structures (i.e. those that are virtually overtwisted), for which a classification scheme is still missing. In this work we use different approaches and employ various techniques to improve our knowledge of symplectic fillings of virtually overtwisted contact structures. We study curves configurations on surfaces to solve the problem in the case of a specific family of lens spaces. Then we give general constraints on the topology …
The infimal Heegaard gradient of a compact 3-manifold was defined and studied by Marc Lackenby in an approach toward the well-known virtually Haken conjecture. As instructive examples, we consider Seifert fibered 3-manifolds, and show that a Seifert fibered 3-manifold has zero infimal Heegaard gradient if and only if it virtually fibers over the circle or over a surface other than the 2-sphere. For a collection of finite coverings of a Seifert fibered 3-manifold, a necessary and sufficient condition to have zero infimal Heegaard gradient is also given.
We use Menke's JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete classification. For virtually overtwisted structures on hyperbolic torus bundles, we show that every strong or exact filling arises from a filling of a tight lens space via round symplectic 1-handle attachment, and we give a condition under which distinct tight lens space filling…
In this article we give a sharp upper bound on the possible values of the Euler characteristic for a minimal symplectic filling of a tight contact structure on a lens space. This estimate is obtained by looking at the topology of the spaces involved, extending this way what we already knew from the universally tight case to the virtually overtwisted one. As a lower bound, we prove that virtually overtwisted structures on lens spaces never bound Stein rational homology balls. Then we turn our attention to covering maps: since an overtwisted disk lifts to an overtwisted disk, all the coverings o…
We investigate great circle links in the three-sphere, the class of links where each component is a great circle. Using the geometry of their complements, we classify such links up to five components. For any two-bridge knot complement, there is a finite cover that is the complement of a link of great circles in $S^3$. We show that for many two-bridge knots, this cover contains a closed incompressible surface. Infinitely many fillings of the two-bridge knot lift to fillings of great circle link where the incompressibility of this surface is preserved. Using this, we show that infinitely many f…
In this paper, we use normal surface theory to study Dehn filling on a knot-manifold. First, it is shown that there is a finite computable set of slopes on the boundary of a knot-manifold that bound normal and almost normal surfaces in a one-vertex triangulation of that knot-manifold. This is combined with existence theorems for normal and almost normal surfaces to construct algorithms to determine precisely which manifolds obtained by Dehn filling: 1) are reducible, 2) contain two--sided incompressible surfaces, 3) are Haken, 4) fiber over the circle, 5) are the 3--sphere, and 6) are a lens s…
We apply representation theory to study the homology of equivariant Dehn-fillings of a given finite, regular cover of a compact 3-manifold with boundary a torus. This yields a polynomial which gives the rank of the part of the homology carried by the solid tori used for Dehn-filling. The polynomial is a symmetrized form of the group determinant studied by Frobenius and Dedekind. As a corollary every such hyperbolic 3-manifold has infinitely many virtually Haken Dehn-fillings.
Let $M$ be a once-punctured torus bundle over $S^1$ with monodromy $h$. We show that, under certain hypotheses on $h$, "most" Dehn-fillings of $M$ (in some cases all but finitely many) are virtually $\mathbb{Z}$-representable. We apply our results to show that surgeries on the figure-eight knot with even numerator are virtually $\mathbb{Z}$-representable.
Hyperbolic Dehn surgery and the bending procedure provide two ways which can be used to describe hyperbolic deformations of a complete hyperbolic structure on a 3-manifold. Moreover, one can obtain examples of non-Haken manifolds without the use of Thurston's Uniformization Theorem. We review these gluing techniques and present a logical continuity between these ideas and gluing methods for Higgs bundles. We demonstrate how one can construct certain model objects in representation varieties $\text{Hom} \left( π_{1} \left( Σ\right), G \right) $ for a topological surface $Σ$ and a semisimple Lie…
Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. Culler and Shalen defined norm on H_1(dM;R) using the SL(2,C) character variety of pi_1(M). The Culler-Shalen norm encodes many topological properties of M; in particular it provides information about Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL_2(C)-character variety of a connected, compact, orientable, irreducible 3-manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a seminorm. The first half of this pap…
This article has two purposes. In \cite{R3} (math.KT/0405211) we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by \cal C) is the key to prove the FIC for 3-manifold groups in general. And we proved the FIC for the fundamental groups of members of a subclass of \cal C. This result was obtained by showing that the double of any member of this subclass is either Seifert fibered or supports a nonpositively curved metric. In this article we prove that for any M in {\cal C} there is a closed 3-manifold P su…
We apply Menke's JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying exact symplectic structures to the …
We prove that an infinite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi-fillings. The argument uses results of Mrowka, Ozsvath and Yu on the translation-invariant solutions to the Seiberg-Witten equations on cylinders and the non-triviality of the Kronheimer-Mrowka monopole invariants of symplectic fillings.
$L^2$-Betti numbers of Dehn fillings
本文研究群论Dehn填充的$L^2$-Betti数，证明了对于一大类virtually special群，其充分深的Dehn填充保持$L^2$-Betti数不变，并应用于验证Singer猜想、建立虚拟纤维化准则及构造具有奇异子群的双曲群新例子。
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